Correlated Brownian Motion and Diffusion of Defects in Spatially Extended Chaotic Systems

Correlated Brownian motion and diffusion of defects in spatially extended chaotic systems

Cite as: Chaos 29, 071104 (2019); https://doi.org/10.1063/1.5113783 Submitted: 07 June 2019 . Accepted: 29 June 2019 . Published Online: 16 July 2019

S. T. da Silva, T. L. Prado, S. R. Lopes , and R. L. Viana

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Chaos 29, 071104 (2019); https://doi.org/10.1063/1.5113783 29, 071104

Correlated Brownian motion and diffusion of defects in spatially extended chaotic systems

Cite as: Chaos 29, 071104 (2019); doi: 10.1063/1.5113783 Submitted: 7 June 2019 · Accepted: 29 June 2019 · Published Online: 16 July 2019 View Online Export Citation CrossMark

S. T. da Silva, T. L. Prado, S. R. Lopes, and R. L. Vianaa)

AFFILIATIONS Departament of Physics, Federal University of Paraná, 81531-990 Curitiba, Paraná, Brazil

a)Author to whom correspondence should be addressed: viana@ﬁsica.ufpr.br

ABSTRACT One of the spatiotemporal patterns exhibited by coupled map lattices with nearest-neighbor coupling is the appearance of chaotic defects, which are spatially localized regions of chaotic dynamics with a particlelike behavior. Chaotic defects display random behavior and diuse along the lattice with a Gaussian signature. In this note, we investigate some dynamical properties of chaotic defects in a lattice of coupled chaotic quadratic maps. Using a recurrence-based diagnostic, we found that the motion of chaotic defects is well-represented by a stochastic time series with a power-law spectrum 1/f σ with 2.3 ≤ σ ≤ 2.4, i.e., a correlated Brownian motion. The correlation exponent corresponds to a memory eect in the Brownian motion and increases with a system parameter as the diusion coecient of chaotic defects. Published under license by AIP Publishing. https://doi.org/10.1063/1.5113783

Recurrence-based diagnostics have been intensively used in recent variety of phenomena have emerged from the interplay between years to unveil dynamical properties of complex dynamical sys- nonlinearity and diusion, such as pattern formation, spatiotempo- tems. Subsets of recurrence plots can be analyzed as microstates, ral intermittency, and turbulence (actually spatiotemporal chaos).7,8 from which a recurrence entropy can be dened in the same way as There is a hierarchy of spatially extended dynamical systems which in information theory. By imposing maximization of this entropy, can be used to investigate spatiotemporal dynamical features, from it is possible to perform a parameter-free recurrence quantica- cellular automata—where space, time, and state variable are all dis- tion analysis. One example is the motion of chaotic defects in crete—to partial dierential equations, for which the same variables coupled map lattices, which is known to share properties of a are continuous.9 One outstanding type of the spatiotemporal sys- Brownian motion like Gaussian diusion. We have used recur- tem is a coupled map lattice, for which the space and time are rence entropy to show that the dynamics of chaotic defects has discrete, but with a continuous state variable.7 Its relative simplicity similar properties with correlated Brownian motion with a power- turns numerical computations easier and faster to perform, and they law spectrum. The memory content of such motion is due to have been extensively used to illustrate many interesting features like the deterministic nature of the causes underlying chaotic defect chimeras.10 motion in a coupled map lattice. We found that the diusion coef- One of the noteworthy features of coupled map lattices is the cient of defects has a dependence on the nonlinearity similar to presence of chaotic defects, dened by Kaneko as a spatially localized the exponent of the Brownian motion power-spectrum. Our anal- chaotic region which separates domains of dierent phases.8,11 More- ysis can be used to investigate other time series with apparently over, the defects themselves are able to move in space, with or without stochastic behavior but possessing an underlying deterministic changes of size. Actually, the motion of a chaotic defect through the mechanism of generation. map lattice resembles a particle undergoing Brownian motion, such that an ensemble defect can diuse. Kaneko has pointed out that, since the coupled map lattice is a deterministic system, it would not be evident that the motion of a defect is really a random walk.11 This Spatiotemporal dynamics is a subject of permanent interest dilemma can be explained by considering that the diusion is trig- in view of existing applications in many research elds, such as gered by the chaotic motion within the defect, i.e., the randomness is hydrodynamics,1 condensed matter physics,2 physical chemistry,3 created by the temporal dynamics of the defect, rather than by global ecology,4 cell biology,5 and neuroscience,6 among others. A great properties of the coupled map lattice.

Chaos 29, 071104 (2019); doi: 10.1063/1.5113783 29, 071104-1 Published under license by AIP Publishing. Chaos ARTICLE scitation.org/journal/cha

In this note, we investigate the chaotic defect motion and argue (a) that it is in fact a Brownian motion with memory, which has similar statistical properties to truly deterministic chaotic systems, and with a power-law spectrum.13 We do so by using a recently developed technique to investigate recurrence-based properties of chaotic systems, which enables us to dene a quantity called recurrence entropy, and which is able to distinguish stochastic systems with memory and deterministic chaotic ones.14 Recurrence plots, which form the basis of this technique, have been extensively used in a variety of applications and provide researchers with a rich toolbox to investigate nonlinear data series, particularly when the data series is short, noise, and nonstationary.15 We use as a paradigmatic model of spatiotemporal dynamics the (b) following coupled map lattice:

ε u(i) = (1 − ε)f (u(i)) + f (u(i+1)) + f (u(i−1)) , (1) n+1 n 2 n n (i) where un is a state variable at discrete time n = 0, 1, 2, ... and discrete position i = 1, 2, ... N in a one-dimensional lattice of size N. The system at each spatial position undergoes a temporal dynamics described by a quadratic map f (u) = (1 + α)u − βu2, with 0 ≤ u ≤ 1, where α and β are system parameters, and ε > 0 is the coupling strength. In the following, we x β = 0.1 and use α as a single variable parameter. The general features of the spatiotemporal dynamics remain practically unchanged if dierent values of 0 < β < 1 are considered. We will consider random initial conditions over the (i) (1) (N) interval 0 ≤ u0 ≤ 1 and xed boundary conditions: un = un = 0. (c) The spatiotemporal dynamics produced by the coupled map lattice (1), for given values of the chosen parameters (ε, α), can be summarized by the phase diagram depicted in Fig. 1(a), which is an isoperiodic diagram showing, for each pair (ε, α), the space-averaged periods of oscillation, which gives us an indication of the dominant temporal period exhibited by the coupled map lattice (since there can be various periodic and chaotic coexisting patterns). For a xed time n , we consider the spatial prole u(i), consisting of various regions 0 n0 with dierent periods of oscillation and dierent spatial lengths. We take the space-average of these periods using the lengths as statistical weights. In this sense, the bluish regions are characterized by mostly low-period attractors (less than 10) or a predominance of ordered patterns. Some blue regions exhibit dominant patterns of well-dened periodicities, as (2, 4, 8) (pattern selection), indicated by 8 label II in Fig. 1(a). Frozen random patterns, for example, are char- FIG. 1. (a) Phase diagram of the coupled map lattice (1): it depicts the domi- acterized by high-period attractors and can be observed in the yellow nant temporal period of the spatiotemporal attractor for given values of ε and α, strips on the left hand side (label I) of Fig. 1(a). for β = 0.1 and N = 100. (b) The density of KS entropy as a function of the In the latter regions of the phase diagram, domains of large same parameters. (c) Space-time plot of the coupled quadratic map lattice (1) for spatial size are usually unstable and domains of short wavelengths ε = 0.1 and α = 2.85. are more common. The shortest domains have wave length 1/2 and are called zigzag patterns. Reddish regions of the phase diagram are related to either dominant high-period or chaotic attractors observed The isoperiodic diagram of Fig. 1(a) is not capable of distin- in the regions labeled III in Fig. 1(a). One example is the chaotic guishing between a high-period and a chaotic attractor. For this defect, which is a spatially localized region of chaotic dynamics which reason, we complement this analysis by computing the Lyapunov N separates zigzag regions of dierent phases. The region of Fig. 1(a) spectrum of the coupled map lattice {λi}i=1. The normalized sum of where chaotic defects show up is contained within the rectangle the positive Lyapunov exponents is a numerical estimate for the den- 2.70 < α < 2.89 and 0.05 < ε < 0.15. Spatiotemporal intermittency sity of Kolmogorov-Sinai (KS) entropy h shown in Fig. 1(b).12 The and fully developed spatiotemporal chaos are labeled as regions IV region in the phase diagram related to the presence of chaotic defects and V, respectively. is marked by a global decrease of the value of h with respect to regions

Chaos 29, 071104 (2019); doi: 10.1063/1.5113783 29, 071104-2 Published under license by AIP Publishing. Chaos ARTICLE scitation.org/journal/cha

diusion observed in an ensemble of such defects. Another char- acterization of the temporal dynamics of a defect is provided by recurrence-based techniques, which have been used in nonlinear data analysis for a long time.16,17 The recurrence plot is a tool to depict recurrences of a K-length time series and is dened as a K × K binary matrix with elements,15

1 if ||xi − xj|| ≤ δ, Rij = i, j = 1, ... , K, (2) (0 if ||xi − xj|| > δ, where δ is the vicinity parameter, || · · · || is an Euclidean norm, and x represents the time series of the position along the x-direction of (d) the center of a defect. Let xn denote the position of the center of a (d) (d) (d) defect at time n. Hence, x = {xn , xn+1, ... , xn+K }, where K denotes the time for which the defect disappears. The most important subsets of R are diagonal lines of “ones” FIG. 2. (a) Motion of an individual defect. (b) Number of chaotic defects as a representing the mutual recurrences of a sequence of points, but function of the lattice size. (c) Mean square displacement of the chaotic defects other subsets of R also have dynamical interpretations like verti- as a function of time for a lattice of N = 6000 maps with α = 2.85, β = ε = 0.1. cal/horizontal lines related to stationary points and the abundance (d) Diffusion coefﬁcient of defects as a function of the nonlinearity parameter α. of isolated points, an indicative of chaotic or stochastic dynamics.15 Recently,14 the concept of subsets of the recurrence plot has been generalized by using recurrence microstates, A(δ), dened as all possible cross-recurrence states among two randomly selected short where spatiotemporal chaos dominate. This global decrease is due to sequences of N points embedded in a K (K Na) length time series the fact that most of the lattice is occupied by zigzag patterns and only (we use Na = 4), such that A(δ) are Na × Na small binary recur- at their boundaries, chaotic defects of limited spatial size are formed. rence matrices obtained according to Eq. (2). For a large enough In particular, for ε = 0.1 and α = 2.85, we observe in the space- randomly selected number of samples M, the recurrence entropy S time plot depicted in Fig. 1(c) the formation of chaotic defects mov- is adequately computed by the Shannon formula,14 ing with time in an irregular manner [Fig. 2(a)]. Two defects can S A = − collide and annihilate mutually or a defect can collide with some ( ) PA ln PA, (3) A boundary and disappear. Due to either reason, it is expected that X the number of defects decay to zero with time. With increasing non- where PA measures the probability of the occurrence of one par- linearity, these defects become larger and their boundaries become ticular microstate A(δ) considering M randomly chosen samples. fuzzier, allowing chaotic motion over a broader spatial range, with Usually, δ is an adjustable parameter as Eq. (3) and A(δ) suggest, alternations between ordered and disordered behavior in both space but this dependence is eliminated observing that S is null when and time (spatiotemporal intermittency). computed for suciently large or small δ.14 So, we impose a natural As shown in Fig. 2(b), the number of chaotic defects increases condition of a maximum for S(δ)18 turning max(S(δ)) ≡ max(S) with the lattice size (at xed time). Since this increase exhibits a lin- and A(δ) ≡ A into parameter-free quantities. The literature shows ear trend, it follows that the density of chaotic defects varies from that max(A) can be used as a parameter-free quantier for time cor- 5% to 10% of the entire lattice. On the other hand, due to collisions related stochastic signals displaying a power spectrum characterized with the boundaries and with themselves, the lifetime of the defects by 1/f σ . Such a feature allows its use to quantify the dynamics of the is expected to decay with time. The average lifetime, on the other appearance of defects over the lattice.14 hand, also varies with the lattice size. Chaotic defects diuse along Figures 3(a)–3(d) depict results for the maximal recurrence the lattice, in a similar manner to Brownian particles, as revealed entropy max(S) computed over the defect signals in the case by Fig. 2(c), where we plot a time series of the mean square dis- α = 2.85, β = ε = 0.1 and also using numerically generated stochas- placement of an ensemble of chaotic defects hδ2i (the lattice with tic signals characterized by a power spectrum proportional to 1/f σ N = 6000 maps has out of ND = 260 defects, over which the aver- for (a) σ = 2.1, (b) σ = 2.3, (c) σ = 2.5, and (d) σ = 2.7. We have age was taken). The data can be tted by the expression hδ2i = 2Dn, used time series of size 30 000 windowed in 300 smaller equally valid for Gaussian diusion, corresponding microscopically to a ran- spaced time series of size N = 1000. The random sampling for each dom walk, where D ≈ 1.25 is a diusion coecient for the defects. windowed time series is M = 105. Comparing values of max(S) Over the range of α-values that present chaotic defects, we found that computed for all signals, we are able to characterize the stochastic their diusion coecient varies between about 1.0 and 1.4, as shown nature of the defect dynamics. in Fig. 2(d). Our results are qualitatively similar to those of Kaneko, As observed in Figs. 3(b) and 3(c), the time evolution of both who considered a dierent map.11 signals is similar, i.e., the defect dynamics can be better charac- From a cursory inspection of Fig. 2(a), the evolution of a chaotic terized by a stochastic signal characterized by a power spectrum defect resembles a Brownian motion of a particle undergoing a ran- following a power law 1/f σ for 2.3 ≤ σ ≤ 2.4. Figures 3(e)–3(h) also dom walk. This observation is further conrmed by the Gaussian show results for max(S) computed for shu ed version of all signals.

Chaos 29, 071104 (2019); doi: 10.1063/1.5113783 29, 071104-3 Published under license by AIP Publishing. Chaos ARTICLE scitation.org/journal/cha

FIG. 3. (a)–(d) Time series of the maximal recurrence entropy max(S) computed for the defect signal (orange lines), superposed with the dynamical behavior of max(S) (blue lines) computed for stochastic time correlated signals following a power-law spectrum 1/f σ , σ = 2.1, 2.3, 2.5, and 2.7, respectively. Blue areas are representative of the stan- dard deviation of each windowed computed max(S). (e)–(h) Shufﬂed version of all signals.

For these cases, the values computed for max(S) are all the same and along the lattice in a particlelike fashion. Since we have used random reproduce the value expected for noncorrelated stochastic signals. initial conditions in numerical simulations, we believe that the for- The value of the power-law exponent σ , characterizing a mation of chaotic defects is quite a general phenomenon. We found stochastic signal with similar maximal entropy evolution, depends on that the number of defects increases with the lattice size and that the value of the nonlinearity parameter α (Fig. 4), which has already defects experience a normal (Gaussian) diusion with a diusion been shown to aect the diusion coecient of defects [see Fig. 2(d)]. coecient around 1.25. This value was found to depend on the non- For the same range of α values considered in the latter gure, we linearity parameter α of the coupled maps. The motion of individual found that the stochastic signals with similar recurrence properties defects is similar to a random walk, and we used a recently developed have exponents lying between 1.8 and 2.4. It seems that a higher value recurrence-based technique to show that it is actually a stochastic of σ is related to the increase of the diusion coecient of defects. We time series displaying a power-law power spectrum f −σ , where σ lies emphasize that varying other system parameters, like β and ε, does between 2.3 and 2.4. The latter exponent was also found to be depen- not aect our results in a signicant way, because we are restricted to dent on the nonlinearity parameter α. In other words, the defect a narrow region in the phase diagram [see Fig. 1(a)] where chaotic motion has recurrence properties similar to a stochastic signal with defects exist. memory. We emphasize that the recurrence entropy technique used In conclusion, we have revisited the formation and dynamics of here is parameter-free, in the sense that it does not depend on ad hoc chaotic defects in lattices of coupled quadratic maps. Chaotic defects sampling parameters. are spatially localized regions of chaotic dynamics which propagate This work was partially supported by the Brazilian Government Agency CNPq (Grant No. 302785/2017-5).

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Chaos 29, 071104 (2019); doi: 10.1063/1.5113783 29, 071104-5 Published under license by AIP Publishing.